2. University of Northern British Columbia, Prince George, Canada Nonlinear and Stochastic Problems in Atmospheric and Oceanic Prediction Workshop Particle filter for data assimilation of nonlinear model systems with non-Gaussian noises Zheqi Shen, Youmin Tang 1. Second Institute of Oceanography, State Oceanic Administration, Hangzhou, China 2. University of Northern British Columbia, Prince George, Canada 2017.11.22

Outline The advantage and challenge of particle filters EnKPF as a hybrid method of EnKF and PF Localization in particle filters

The advantages and challenges of particle filter 01 The advantages and challenges of particle filter

Prediction /Simulation Data assimilation Numerical models ICs Parameters 𝛼=？ 𝜅=？ Prediction /Simulation Data assimilation methods Observation data

Data assimilation methods Variational methods Data assimilation methods Ensemble Kalman filters Ensemble-based filters Particle filters

Ensemble Kalman filter 𝑋 𝑘+1 = 𝑓(𝑋 𝑘 )+ 𝜁 𝑘 𝑦 𝑘 =ℎ( 𝑋 𝑘 ) Ensemble Adjustment Kalman filter State space model Compute the mean and covariance of the forecast state using the ensemble Compute the posterior mean and covariance of the analysis state by Gaussian assumption Adjust the ensemble to adapt the analysis mean and covariance 1 2 3

Particle filter Particle filter Using observations to update the weights, using weighted mean to give analysis Resampling is frequently used to redistribute the ensemble according to the weights The main challenge of PF is the filter degeneracy.

Comparison of PF and EnKF Lorenz ’63 model: 𝑑𝑥 𝑑𝑡 =𝜎 𝑦−𝑥 𝑑𝑦 𝑑𝑡 =𝜌𝑥−𝑦−𝑥𝑧 𝑑𝑧 𝑑𝑡 =𝑥𝑦−𝛽𝑧 No Gaussian assumptions Weights to approximate the full probability density functions (PDFs)

Filter degeneracy 256 members EAKF 256 members PF 32 members PF The effective ensemble size 𝑁 𝑒𝑓𝑓 =1/ 𝑖=1 𝑁 𝑤 𝑖 2 , which measures the ensemble diversity. 𝑁 𝑒𝑓𝑓 =𝑁, if each 𝑤 𝑖 =1/𝑁 (in EAKF). 𝑁 𝑒𝑓𝑓 will become very small when very few members have weights, that leads degeneracy.

02 Our first attempt to overcome the filter degeneracy - Ensemble Kalman particle filter

A hybrid method of EnKF and PF Ensemble Kalman particle filter (Frei and Kunsch, 2013) Modifying the ensemble members Computing weights and resampling In one analysis step PF

Ensemble Kalman particle filter (EnKPF) 𝑥 𝑖 𝑢 = 𝑥 𝑖 𝑓 + 𝐾 1 𝑦−ℎ 𝑥 𝑖 𝑓 𝐾 1 = 𝑃 𝑓 𝐻 𝑇 𝐻 𝑃 𝑓 𝐻 𝑇 + 𝑅 𝛾 −1 -- EnKF with inflated observational error 𝑄= 𝐾 1 𝑅 𝐾 1 𝑇 /𝛾 𝑤 𝑖 = exp 𝑦−ℎ 𝑥 𝑖 𝑢 𝑇 𝑅 1−𝛾 +𝐻𝑄 𝐻 𝑇 −1 𝑦−ℎ 𝑥 𝑖 𝑢 , w i = w i 𝑤 𝑖 -- computed the weights according to the modified likelihood function Resampling : 𝑥 𝑖 𝑣 = 𝑥 𝑠(𝑖) 𝑢 +𝜖 where 𝑠 𝑖 is the resampling index 𝑥 𝑖 𝑎 = 𝑥 𝑖 𝑣 + 𝐾 2 𝑦+ 𝜖 ′ −ℎ 𝑥 𝑖 𝑣 𝐾 2 =𝑄 𝐻 𝑇 𝐻𝑄 𝐻 𝑇 + 𝑅 1−𝛾 −1 -- update the resampled ensemble In EnKF 𝐾= 𝑃 𝑓 𝐻 𝑇 𝐻 𝑃 𝑓 𝐻 𝑇 +𝑅 −1 In PF 𝑤 𝑖 ∝exp{ 𝑦−ℎ 𝑥 𝑓 𝑇 𝑅 −1 {[𝑦−ℎ 𝑥 𝑓 ]} The ‘progress correction’ idea (Musso et al., 2001) (Frei and Kunsch, 2013)

An extension of EnKPF for nonlinear h 𝑥 𝑖 𝑢 = 𝑥 𝑖 𝑓 + 𝐾 1 𝑦−ℎ 𝑥 𝑖 𝑓 𝐾 1 = 𝑃 𝑓 𝐻 𝑇 𝐻 𝑃 𝑓 𝐻 𝑇 + 𝑅 𝛾 −1 -- EnKF with inflated observational error 𝑄= 𝐾 1 𝑅 𝐾 1 𝑇 /𝛾 𝑤 𝑖 = exp 𝑦−ℎ 𝑥 𝑖 𝑢 𝑇 𝑅 1−𝛾 +𝐻𝑄 𝐻 𝑇 −1 𝑦−ℎ 𝑥 𝑖 𝑢 , w i = w i 𝑤 𝑖 -- computed the weights according to the modified likelihood function Resampling : 𝑥 𝑖 𝑣 = 𝑥 𝑠(𝑖) 𝑢 +𝜖 where 𝑠 𝑖 is the resampling index 𝑥 𝑖 𝑎 = 𝑥 𝑖 𝑣 + 𝐾 2 𝑦+ 𝜖 ′ −ℎ 𝑥 𝑖 𝑣 𝐾 2 =𝑄 𝐻 𝑇 𝐻𝑄 𝐻 𝑇 + 𝑅 1−𝛾 −1 -- update the resampled ensemble When the measurement function is nonlinear 𝑦=ℎ(𝑥) nEnKPF (Frei and Kunsch, 2013) mEnKPF

Adaptive algorithm to determine 𝛾 Target of the algorithm is to maintain the effective ensemble size 𝑁 𝑒𝑓𝑓 into a prescribe interval by adjusting 𝛾 Initial guess of 𝛾 (typically, 1/2) Apply EnKF and PF with the parameter 𝛾 𝛾=𝛾+ 1 2 𝑘 𝛾=𝛾− 1 2 𝑘 Compute the 𝑁 𝑒𝑓𝑓 according to the weights no Is 𝑁 𝑒𝑓𝑓 in the interval 𝑁∗[ 𝜏 1 , 𝜏 2 ]? yes End the loop

EnKPF in Lorenz models Lorenz ’63 model: 𝑑𝑥 𝑑𝑡 =𝜎 𝑦−𝑥 𝑑𝑦 𝑑𝑡 =𝜌𝑥−𝑦−𝑥𝑧 𝑑𝑧 𝑑𝑡 =𝑥𝑦−𝛽𝑧 Nonlinear measurement: 𝑦=5tanh⁡(𝑥) N = 64

EnKPF in Lorenz models Different nonlinear measurement functions 𝑑 𝑋 𝑗 𝑑𝑡 = 𝑋 𝑗+1 − 𝑋 𝑗−2 𝑋 𝑗−1 − 𝑋 𝑗 +𝐹 𝑗=1,2,…,𝐽 𝑦=5tanh⁡(𝑥) or 𝑦=5tanh⁡(2𝜋𝑥) Different nonlinear measurement functions Different data frequencies (nonlinearity of the model)

Conclusions for EnKPF EnKPF combines the merits of EnKF and SIR-PF, which can assimilate non- Gaussian information with a relatively small ensemble. The EnKPF algorithm can be extended to cases that the measurement function is nonlinear. Regarding the implicit linearization in nonlinear h, we have proposed two EnKPF schemes, namely, nEnKPF and mEnKPF. In most cases, mEnKPF performs better than nEnKPF, both outperform EnKF. References Frei, M., & Künsch, H. R. (2013). Bridging the ensemble Kalman and particle filters. Biometrika, 100(4), 781-800. Shen, Z., & Tang, Y. (2015). A modified ensemble Kalman particle filter for non‐Gaussian systems with nonlinear measurement functions. Journal of Advances in Modeling Earth Systems, 7(1), 50 - 66. Shen, Z., Zhang, X., & Tang, Y. (2016). Comparison and combination of EAKF and SIR-PF in the Bayesian filter framework. Acta Oceanologica Sinica, 35(3), 69-78. doi: 10.1007/s13131-015-0757-x

03 A more practical strategy to avoid filter degeneracy - Localization in particle filters

Filter degeneracy and scalar weights 𝒘 𝒏 ∝ 𝒊=𝟏 𝒎 𝒆𝒙𝒑⁡{− 𝟏 𝟐 𝒚 𝒊 − 𝒉 𝒊 𝑿 𝒏 𝟐 / 𝝈 𝒊 𝟐 } Traditional PF uses scalar weights The effect of each observation is accumulated into a single number as a global weight for each particle Filter degeneracy makes traditional PF invalid with real models While the dimension of the dynamical model and the number of observations is large, it is very likely to get filter degeneracy

From scalar weights to vector weights 𝒘 𝒏 ∝ 𝒊=𝟏 𝒎 𝒆𝒙 𝒑 − 𝟏 𝟐 𝒚 𝒊 − 𝒉 𝒊 𝑿 𝒏 𝟐 𝝈 𝒊 𝟐 = 𝒊=𝟏 𝒎 𝑝 𝑦 𝑖 𝑋 𝑛 Extending to vector weights 𝑤 1 𝑛 𝑤 2 𝑛 𝑤 𝑗 𝑛 𝒍 𝒊𝒋 = − 𝟏 𝟒 𝒅 𝒊𝒋 𝒄 𝟓 + 𝟏 𝟐 𝒅 𝒊𝒋 𝒄 𝟒 + 𝟓 𝟖 𝒅 𝒊𝒋 𝒄 𝟑 − 𝟓 𝟑 𝒅 𝒊𝒋 𝒄 𝟐 +𝟏, 𝟎< 𝒅 𝒊𝒋 <𝒄 𝟏 𝟏𝟐 𝒅 𝒊𝒋 𝒄 𝟓 − 𝟏 𝟐 𝒅 𝒊𝒋 𝒄 𝟒 + 𝟓 𝟖 𝒅 𝒊𝒋 𝒄 𝟑 + 𝟓 𝟑 𝒅 𝒊𝒋 𝒄 𝟐 −𝟓 𝒅 𝒊𝒋 𝒄 +𝟒− 𝟐 𝟑 𝒅 𝒊𝒋 𝒄 −𝟏 , 𝒄< 𝒅 𝒊𝒋 <𝟐𝒄 𝟎, 𝒅 𝒊𝒋 >𝟐𝒄 Poterjoy (2016) 𝑤 𝑗 𝑛 = 𝑖=1 𝑚 [𝑝( 𝑦 𝑖 | 𝑋 𝑛 ) 𝑙 𝑖𝑗 +1∗(1− 𝑙 𝑖𝑗 )] Shen et al. (2017) 𝑤 𝑗 𝑛 =exp {− 1 2 𝑖=1 𝑚 𝑙 𝑖𝑗 [ 𝑦 𝑖 − ℎ 𝑖 ( 𝑋 𝑛 ) ] 2 / 𝜎 𝑖 2 }.

𝑤 𝑗 𝑛 ∝ 𝑖=1 𝑚 [𝑝( 𝑦 𝑖 | 𝑋 𝑛 ) 𝑙 𝑖𝑗 +(1− 𝑙 𝑖𝑗 )] Poterjoy (2016) Shen et al. (2017) 𝑤 𝑗 𝑛 ∝ 𝑖=1 𝑚 [𝑝( 𝑦 𝑖 | 𝑋 𝑛 ) 𝑙 𝑖𝑗 +(1− 𝑙 𝑖𝑗 )] 𝑤 𝑗 𝑛 ∝exp {− 1 2 𝑖=1 𝑚 𝑙 𝑖𝑗 [ 𝑦 𝑖 − ℎ 𝑖 ( 𝑋 𝑛 ) ] 2 / 𝜎 𝑖 2 }. 𝑤 1 𝑛 𝑤 2 𝑛 𝑤 1 𝑛 𝑤 2 𝑛 𝑤 𝑗 𝑛 𝑤 𝑗 𝑛 𝒑 𝒚 𝒊 𝑿 𝒏 = 𝒆𝒙𝒑⁡{− 𝟏 𝟐 𝒚 𝒊 − 𝒉 𝒊 𝑿 𝒏 𝟐 / 𝝈 𝒊 𝟐 } 𝑝( 𝑦 2 | 𝑋 𝑛 ) 𝑝( 𝑦 1 | 𝑋 𝑛 )

Resampling for vector weights The problem of discontinuity for vector weights Merging the resampled and original particles 𝒙 𝒊 = 𝒙 𝒊 + 𝒓 𝟏 𝒙 𝒊 𝒓𝒆𝒔𝒂𝒎𝒑 − 𝒙 𝒊 + 𝒓 𝟐 ( 𝒙 𝒊 − 𝒙 𝒊 )

Comparison with P2016 LPF and EAKF Lorenz ‘96 model 36 variables Nonlinear observation y=|x| Accuracy and consistency ensemble size is 80

With nonlinear measurement functions Different nonlinear measurement functions Different ensemble sizes The advantage is even more significant when the nonlinearity of h is strong

Conclusions for LPF We have improved the formula to generate vector weights, the new LPF outperforms both EAKF and P2016 LPF in twin-experiments with nonlinear measurement functions LPF can avoid filter degeneracy with affordable number of particles (computational resources), has a great potential to be applied in real models. References Poterjoy, J. (2016). A localized particle filter for high-dimensional nonlinear systems. Monthly weather review, 144(1), 59-76. Shen, Z., Tang, Y., & Li, X. (2017). A new formulation of vector weights in localized particle filter. Quarterly Journal of the Royal Meteorological Society. doi: 10.1002/qj.3180

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